Fano fourfold literature

an overview of the literature listing Fano fourfolds

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Pas25 Casagrande–Druel varieties with Lefschetz defect 2

Pastorino, F. (2025). On Casagrande-Druel Fano varieties with Lefschetz defect 2.

count147 families
constructioncasagrande-druel
Picard rank $\rho_X$$\ge 4$
Lefschetz defect $\delta_X$2
entry numberingfamily number
invariantshodge, numerical, lefschetz_defect, construction
linksjournal arXiv:2510.20732
notes147 distinct families with ρ_X ≥ 4 and Lefschetz defect δ_X = 2, classified via Construction A. Numerical invariants and Hodge numbers provided. arXiv preprint (no MR yet).

Varieties

Each row is a family from this source. The description column records the construction as given in the paper (ambient and defining bundle, Cox-ring data, base variety, or birational model, depending on the source); the invariant columns are those the paper tabulates. Provenance and conventions are noted below the table.

How to read the construction. Each family is built by Casagrande and Druel's Construction A from a Fano threefold $Z$, whose Mori–Mukai label $\#a\text{-}b$ is given in the description. Construction A takes a suitable $\mathbb{P}^1$-bundle over $Z$ and blows it up along two disjoint sections; the result is a Fano fourfold with Lefschetz defect $\delta_X=2$. This is why every entry shares $\delta_X=2$ and records which threefold $Z$ (and, where relevant, which product or toric model) it comes from.

entry$\rho_X$$\delta_X$$(-\mathrm{K}_X)^4$$\mathrm{h}^0(-\mathrm{K}_X)$$\mathrm{h}^{1,1}$$\mathrm{h}^{1,2}$$\mathrm{h}^{1,3}$$\mathrm{h}^{2,2}$description
Pas25:rho4-1421122942014from Z=#2-18
Pas25:rho4-2421423541014from Z=#2-25
Pas25:rho4-3421483640012from Z=#2-24
Pas25:rho4-4421523742010from Z=#2-18
Pas25:rho4-542164394206from Z=#2-25
Pas25:rho4-6421724140014from Z=#2-29
Pas25:rho4-7421724240342from Z=#2-34
Pas25:rho4-842176414206from Z=#2-25
Pas25:rho4-9421784240012from Z=#2-27
Pas25:rho4-10421804340124from Z=#2-32
Pas25:rho4-1142184444016from Z=#2-33
Pas25:rho4-12421904540021from Z=#2-34
Pas25:rho4-1342194454009from Z=#2-24
Pas25:rho4-1442199464109from Z=#2-25
Pas25:rho4-15422054840124from Z=#2-34
Pas25:rho4-16422084840012from Z=#2-31
Pas25:rho4-1742208484306from Z=#3-24
Pas25:rho4-1842208484306from Z=#3-24, product (#3-9)xP^1
Pas25:rho4-19422104940124from Z=#2-35
Pas25:rho4-2042216494206from Z=#2-25
Pas25:rho4-21422185040012from Z=#2-32
Pas25:rho4-2242221544008from Z=#2-27
Pas25:rho4-23422235140014from Z=#2-34
Pas25:rho4-24422305240010from Z=#2-30
Pas25:rho4-25422335340014from Z=#2-33
Pas25:rho4-26422385440014from Z=#2-34
Pas25:rho4-27422385440012from Z=#2-34
Pas25:rho4-2842241504006from Z=#2-27
Pas25:rho4-2942241544106from Z=#2-34
Pas25:rho4-3042246554008from Z=#2-29
Pas25:rho4-31422485640012from Z=#2-34
Pas25:rho4-32422485640012from Z=#2-35
Pas25:rho4-3342256574006from Z=#2-32
Pas25:rho4-3442256574008from Z=#2-32
Pas25:rho4-3542256574106from Z=#2-34, product (#3-14)xP^1
Pas25:rho4-36422655940011from Z=#2-35
Pas25:rho4-3742266594006from Z=#2-32
Pas25:rho4-3842266594009from Z=#2-34
Pas25:rho4-39422686040012from Z=#2-36
Pas25:rho4-4042271604106from Z=#2-34
Pas25:rho4-4142271604009from Z=#2-34
Pas25:rho4-4242281624006from Z=#2-34
Pas25:rho4-4342282624008from Z=#2-32
Pas25:rho4-4442282624008from Z=#2-33
Pas25:rho4-4542284624006from Z=#2-34
Pas25:rho4-4642286634006from Z=#2-35
Pas25:rho4-4742288634007from Z=#2-31
Pas25:rho4-4842296654006from Z=#2-35
Pas25:rho4-4942299654007from Z=#2-30
Pas25:rho4-5042301664006from Z=#2-34
Pas25:rho4-5142302664009from Z=#2-34
Pas25:rho4-5242303664007from Z=#2-35
Pas25:rho4-5342304664006from Z=#2-32
Pas25:rho4-5442304664006from Z=#2-34, product (#3-19)xP^1
Pas25:rho4-5542304664006from Z=#2-34
Pas25:rho4-5642308674008from Z=#2-33
Pas25:rho4-5742314684006from Z=#2-34
Pas25:rho4-5842316684006from Z=#2-30
Pas25:rho4-5942319694006from Z=#2-34
Pas25:rho4-6042320694006from Z=#2-32
Pas25:rho4-6142320694006from Z=#2-34, product (#3-22)xP^1
Pas25:rho4-6242320694006from Z=#2-35
Pas25:rho4-6342329714007from Z=#2-35
Pas25:rho4-6442331714006from Z=#2-31
Pas25:rho4-6542335724006from Z=#2-34
Pas25:rho4-6642337724006from Z=#2-34, toric (I_{15})
Pas25:rho4-6742346744006from Z=#2-35
Pas25:rho4-6842347844006from Z=#2-34, toric (I_{12})
Pas25:rho4-6942351754005from Z=#2-35, toric (H_{10})
Pas25:rho4-7042356764106from Z=#2-29
Pas25:rho4-7142356764006from Z=#2-34
Pas25:rho4-7242357764006from Z=#2-33, toric (I_{14})
Pas25:rho4-7342362774006from Z=#2-33
Pas25:rho4-7442367784005from Z=#2-34, #2-35, toric (H_{9})
Pas25:rho4-7542368784006from Z=#2-31
Pas25:rho4-7642368784006from Z=#2-34, product (#3-26)xP^1, toric (I_{13})
Pas25:rho4-7742378804005from Z=#2-34, product P^2xBl_2P^2, toric (H_{8})
Pas25:rho4-7842382814005from Z=#2-34, #2-36, toric (H_{7})
Pas25:rho4-7942382814006from Z=#2-34
Pas25:rho4-8042384814006from Z=#2-35, toric (I_{8})
Pas25:rho4-8142389824006from Z=#2-34, toric (I_{10})
Pas25:rho4-8242390624006from Z=#2-33, toric (I_{9})
Pas25:rho4-8342400844006from Z=#2-34, product (#3-29)xP^1, toric (I_{7})
Pas25:rho4-8442409864005from Z=#2-35, toric (H_{6})
Pas25:rho4-8542411864006from Z=#2-33, toric (I_{6})
Pas25:rho4-8642414874106from Z=#2-29
Pas25:rho4-8742415874006from Z=#2-33, toric (I_{5})
Pas25:rho4-8842415874005from Z=#2-34, #2-35, toric (H_{5})
Pas25:rho4-8942433904006from Z=#2-33, toric (I_{4})
Pas25:rho4-9042442924006from Z=#2-35, toric (I_{3})
Pas25:rho4-9142447934005from Z=#2-35, toric (H_{4})
Pas25:rho4-9242463964006from Z=#2-34, toric (I_{2})
Pas25:rho4-9342478994005from Z=#2-34, #2-36, toric (H_{3})
Pas25:rho4-94424961024006from Z=#2-33, toric (I_{1})
Pas25:rho4-95425051044005from Z=#2-36, toric (H_{2})
Pas25:rho4-96425581144005from Z=#2-36, toric (H_1)
Pas25:rho5-1521804350126from Z=#3-27
Pas25:rho5-2521844350012from Z=#3-17
Pas25:rho5-3522004650012from Z=#3-19
Pas25:rho5-4522024750016from Z=#3-28
Pas25:rho5-5522024750016from Z=#3-27
Pas25:rho5-6522144950012from Z=#3-25
Pas25:rho5-7522205050010from Z=#3-24
Pas25:rho5-852224515108from Z=#3-27
Pas25:rho5-952224515108from Z=#3-27, product (#4-2)xP^1
Pas25:rho5-10522245150012from Z=#3-27
Pas25:rho5-11522295250012from Z=#3-28
Pas25:rho5-12522345350012from Z=#3-27
Pas25:rho5-13522365350010from Z=#3-17
Pas25:rho5-14522445550012from Z=#3-31
Pas25:rho5-15522455550010from Z=#3-28
Pas25:rho5-1652246555008from Z=#3-27
Pas25:rho5-17522505650010from Z=#3-30
Pas25:rho5-18522525650010from Z=#3-26, #3-19
Pas25:rho5-1952256575008from Z=#3-28
Pas25:rho5-2052256575008from Z=#3-28, product (#4-4)xP^1
Pas25:rho5-2152256575008from Z=#3-27, product (#4-5)xP^1
Pas25:rho5-22522565750010from Z=#3-27
Pas25:rho5-2352266595008from Z=#3-27
Pas25:rho5-2452278615008from Z=#3-27
Pas25:rho5-2552278615009from Z=#3-24
Pas25:rho5-26522826250010from Z=#3-27
Pas25:rho5-2752283625009from Z=#3-25
Pas25:rho5-2852288635008from Z=#3-27, product (#4-7)xP^1
Pas25:rho5-2952288635008from Z=#3-27
Pas25:rho5-3052293645009from Z=#3-28
Pas25:rho5-3152299715008from Z=#3-28, toric (Q_{17})
Pas25:rho5-3252304665008from Z=#3-27, product (#4-8)xP^1
Pas25:rho5-3352310675008from Z=#3-27, #3-25, toric (Q_{16})
Pas25:rho5-3452310675009from Z=#3-26, toric (Yes^3 note{a})
Pas25:rho5-3552320785008from Z=#3-28, #3-27, product (#4-9)xP^1, toric (Q_{15})
Pas25:rho5-3652325705008from Z=#3-30, #3-28, toric (Q_{14})
Pas25:rho5-3752330715008from Z=#3-31, #3-27, toric (Q_{13})
Pas25:rho5-3852330715008from Z=#3-27
Pas25:rho5-3952331715008from Z=#3-28, #3-25, toric (Q_{12})
Pas25:rho5-4052336725008from Z=#3-28, product F_1xBl_2P^2, toric (Q_{10})
Pas25:rho5-4152336725008from Z=#3-27, product (#4-10)xP^1, toric (Q_{11})
Pas25:rho5-4252341735008from Z=#3-28, toric (Q_{9})
Pas25:rho5-4352352755008from Z=#3-28, #3-27, product (#4-11)xP^1, toric (Q_{8})
Pas25:rho5-4452363775008from Z=#3-28, toric (Q_{7})
Pas25:rho5-4552368785008from Z=#3-28, product (#4-12)xP^1, toric (Q_{6})
Pas25:rho5-4652373795008from Z=#3-30, #3-28, toric (Q_{5})
Pas25:rho5-4752394835008from Z=#3-31, #3-27, toric (Q_{3})
Pas25:rho5-4852405855008from Z=#3-31, #3-28, toric (Q_{4})
Pas25:rho5-4952405855008from Z=#3-30, toric (Q_2)
Pas25:rho5-5052442925008from Z=#3-31, toric (Q_1)
Pas25:rho6-1626Bl_2P^2 x Bl_2P^2 (the unique rho=6 family; no Hodge row in the paper)

Invariants from Appendix A of arXiv:2510.20732. volume = (-K)^4, anticanonical = h^0(-K), h11 = Picard rank.